Walras and dividends equilibrium with possibly satiated consumers

Walras and dividends equilibrium with possibly satiated consumers Nizar Allouch Department of Economics Queen Mary, University of London Mile End Rd, E1 4NS London, United Kingdom n.allouch@qmul.ac.uk Cuong Le Van Centre d Economie de la Sorbonne Paris 1 Pantheon-Sorbonne, CNRS 106-112 Bd de l Hôpital 75647 Paris, France levan@univ-paris1.fr Current Version: January, 2006 Abstract The main contribution of the paper is to provide a weaker nonsatiation assumption than the one commonly used in the literature to ensure the existence of competitive equilibrium. Our assumption allows for satiation points inside the set of individually feasible consumptions, provided that the consumer has satiation points available to him outside this set. As a result, we show the concept of equilibrium with dividends (See Aumann and Dreze (1986), Mas-Collel (1992)) is pertinent only when the set of satiation points is included in the set of individually feasible consumptions. Our economic motivation stems from the fact that in decentralized markets, increasing the incomes of consumers through dividends, if it is possible, is costly since it involves the intervention of a social planner. Then, we show, in particular, how in securities markets our weak nonsatiation assumption is satisfied by Werner s (1987) assumption. JEL classification codes: D51, C71. Keywords: Satiation, Dividends, Equilibrium, Exchange Economy, Shortselling. The authors thank Bernard Cornet for helpful comments. 1

1 Introduction Since the seminal contributions of Arrow-Debreu (1954), and Mckenzie (1959), on the existence of a competitive equilibrium, a subject of ongoing interest in the economics profession has been the robustness of the various assumptions made to ensure such a result. On the consumer side, assumptions such as the convexity of preferences, free-disposal and survival have been investigated both conceptually and empirically by numerous economists ranging from development economists to decision theorists. The seemingly innocuous assumption of nonsatiation, normally represented in Microeconomics textbooks by the monotonicity of preferences, appears to have received much less attention. Perhaps, the main critique to the insatiability assumption is that the human nature calls it into question. Namely, any moderately greedy person will testify to their occasional satiation. Technically, a satiation point seems to be genuinely guaranteed with continuous preferences, whenever the choice set is bounded. Having a bounded choice set is hardly surprising, as consumption activities take place over a limited time span. Accordingly, this condition has been weakened by assuming that nonsatiation holds only over individually feasible consumptions 1 ; that is to say, satiation levels are higher than the actual consumption levels involved in trade. In the presence of satiation points in individually feasible consumption sets, we find in the literature the concept of equilibrium with coupons or dividends that extend the classical general equilibrium theory to the class of such economies (see Aumann and Dreze (1986), Mas-Collel (1992), Kaji (1996), Cornet, Topuzu and Yildiz (2003)) 2. The underlying idea is to allow the nonsatiated consumers to benefit, through dividends, from the budget surplus created by non budget-binding optimal consumptions of satiated consumers. The analysis of the above-named authors has proved to be relevant to the study of markets with price rigidities, such as Labor market. One issue with equilibrium with dividends is that, increasing the incomes of consumers in decentralized markets, if it is possible, is costly since it involves the intervention of a social planner. In this paper, our main contribution is to introduce a weak nonsatiation 1 Individually feasible consumptions are defined as the projection, on individual s choice set, of consumptions that could potentially be achieved by trade 2 See also Dreze and Muller (1980), Florig and Yildiz (2002), Le Van and Minh (2004), Konovalov (2005). 2

assumption that ensures the existence of an exact competitive equilibrium (without dividends). Our assumption allows for the satiation points in individually feasible consumption set, provided that the satiation area is not a subset of the individually feasible consumptions. Stated formally, if we consider M i to be the maximum of the utility function of consumer i over individually feasible consumption set, our assumption stipulates that there be a consumption bundle outside the individually feasible consumption set that guarantees at least the utility level M i. The standard nonsatiation assumption rather requires that there be a consumption bundle outside the individually feasible consumption set that guarantees strictly more than the utility level M i. In a recent paper, Won and Yannelis (2005) demonstrate the existence of a competitive equilibrium with a different nonsatiation assumption, in a more general setting. Their assumption allows the satiation area to be inside the individually feasible consumption sets, provided that it is unaffordable with respect to any price system supporting the preferences of the nonsatiated consumers. Won and Yannelis (2005) also show that their assumption is implied by our weak nonsatiation assumption and could be suitably applied to some asset pricing models. Notwithstanding the novelty of their approach, their assumption relies on price systems, whereas our weak nonsatiation is defined on the primitives of the economy. In securities markets with short-selling, Werner (1987) introduces a nonsatiation assumption which allows the existence of satiation points even if they are in the projections of the feasible set. Werner s assumption stipulates that each trader has a useful portfolio. This is defined as a portfolio which, when added, at any rate, to any given portfolio increases the trader s utility. In particular his assumption implies an unbounded set of satiation points. In the paper, we show that Werner s nonsatiation implies our weak nonsatiation assumption. We also provide an example where Werner s nonsatiation does not hold, whereas our weak nonsatiation assumption is satisfied, and consequently an equilibrium exists. The paper is organized as follows, Section 2 is devoted to the model. In Section 3, we shall introduce our new nonsatiation assumption. In Section 4, we compare our new nonsatiation assumption with Werner s nonsatiation. Section 5 is an Appendix. 3

2 The Model We consider an economy with a finite number l of goods and a finite number I of consumers. For each i I, let X i R l denote the set of consumption goods, let u i : X i R denote the utility function and let e i X i be the initial endowment. In the sequel, we will denote this economy by E = {(X i, u i, e i ) i I }. An individually rational feasible allocation is the list (x i ) i I i I X i, which satisfies i I x i = i I e i, and u i (x i ) u i (e i ), i I. We denote by A the set of individually rational feasible allocations. We shall denote by A i the projection of A onto X i. The set of individually rational utilities is given by U = {(v i ) R I + there exists x A s.t. u i (e i ) v i u i (x i ), i I}. In the following, for simplicity, U will be called utility set. We consider the following definition of Walras equilibrium (resp. quasiequilibrium). Definition 1. A Walras equilibrium (resp. quasi-equilibrium) of E is a list ((x i ) i I, p ) i I X i (R l \ {0}) which satisfies: (a) i I x i = i I e i (Market clearing); (b) for each i one has p x i = p e i (Budget constraint), and for each x i X i, with u i (x i ) > u i (x i ), it holds p x i > p e i. [resp. p x i p e i ]. In the presence of satiation points in individually feasible consumption sets, we find in the literature the concept of equilibrium with dividends (see Aumann and Dreze (1986), Mas-Collel (1992), Cornet, Topuzu and Yildiz (2003)). The dividends increase the income of nonsatiated consumers, in order to capture the surplus created by satiated consumers. In the following, we define the concept of equilibrium (resp. quasi-equilibrium) with dividends. Definition 2. An equilibrium (resp. quasi-equilibrium) with dividends (d i ) i I R I + of E is a list ((x i ) i I, p ) i I X i R l which satisfies: (a) i I x i = i I e i (Market clearing); (b) for each i I one has p x i p e i + d i (Budget constraint), and for each x i X i, with u i (x i ) > u i (x i ), it holds p x i > p e i + d i. [resp. p x i p e i + d i ]. 4

When d i = 0, for all i I, an equilibrium (resp. quasi-equilibrium) with dividends is a Walras equilibrium (resp. quasi-equilibrium). Remark The passage from a quasi-equilibrium with dividends to an equilibrium with dividends is similar to the one in the standard Walrasian case. That is to say, let ((x i ) i I, p ) be a quasi-equilibrium with dividends (d i ) i I. Assume that for all i I, the set {x i X i u i (x i ) > u i (x i )} is relatively open in X i, and inf p X i < p x i, then, ((x i ) i I, p ) is an equilibrium with dividends (d i ) i I. Now, we list our assumptions: (H 1 ) For each i I, the set X i is closed and convex. (H 2 ) For each i I, the utility function u i is strictly quasi-concave and upper semicontinuous 3. (H 3 ) The utility set U is compact. (H 4 ) For each i I, for all x i A i, there exists x i X i such that u i (x i) > u i (x i ). For every i I, let S i = {x i X i : u i (x i ) = max xi X i u i (x i )}. The set S i is the set of satiation points for agent i. Observe that under assumptions (H1) (H2), the set S i is closed and convex. 3 The Results 3.1 Equilibrium with dividends We first give an existence of Walras quasi-equilibrium theorem when there exists no satiation. Theorem 1. Assume (H 1 ) (H 4 ), then there exists a Walras quasi-equilibrium. Proof. The proof is quite standard. See e.g. Arrow and Debreu (1954) when the consumption sets are bounded from below, or the proof given in Dana, Le Van and Magnien (1999) for an exchange economy. 3 We recall that a function u i is said to be quasi-concave if its level-set L α = {x i X i : u i (x i ) α} is convex, for each α R. The function u i is strictly quasi-concave if and only if, for all x i, x i X i, u i (x i ) > u i(x i ) and λ [0, 1), then u i (λx i + (1 λ)x i ) > u i(x i ). It means that u i (λx i + (1 λ)x i ) > min(u i (x i ), u i (x i )), if u i(x i ) u i (x i ). The function u i is upper semicontinuous if and only if L α is closed for each α. 5

Now, we come to our first result. This result has been proved by adding a virtual commodity to the economy and then modifying the utility functions of the agents. Our proof follows the steps of Le Van and Minh (2004) in introducing a new commodity, but our modification of the utility functions differs from theirs. The advantage of such a modification will become clear when we introduce a new nonsatiation assumption. Let us recall the following definition. Definition 3. Let B be a closed convex nonempty set of R l, where l is an integer. The recession cone of B, denoted by O + B, is defined as follows: O + B = {w R l : x B, λ 0, x + λw B.} We first give an intermediate result. The proof of the result is new, since we use a new modification of utility functions. The modified economy is then used to establish the existence of a quasi-equilibrium with dividends for the initial economy. In the following, we restrict the economy to compact consumption sets. Proposition 1. Assume (H 1 ) (H 2 ), and for every i I, the consumption set X i is compact. Then there exists a quasi-equilibrium with dividends. Proof. Let us introduce the auxiliary economy Ê = { ( X i, û i, ê i ) i I }, where X i = X i R +, ê i = (e i, δ i ) with δ i > 0 for any i I and the utility functions û i are defined as follows. Let M i = max {u i (x i ) x i A i }. Case 1. If there exists x i A c i (the complement of A i in X i ), such that u i (x i ) > M i, then û i (x i, d i ) = u i (x i ) for every (x i, d i ) X i. Case 2. Now, consider the case where there exists no x i A c i which satisfies u i (x i ) > M i, but there exists x i A c i, such that u i (x i ) = M i. We modify agent s i utility function as follows: Using x i we define the function λ i ( ) : S i R + {+ }, where, λ i (x i ) = sup{β R + x i + β(x i x i ) S i }. 6

Now, using the function λ i, we can define a new utility function, û i, for agent i: { ui (x i ) + 1 1 λ û i (x i, d i ) =, if x i (x i ) i S i u i (x i ), otherwise, for every (x i, d i ) X i. Case 3. If there exists no x i A c i, such that u i (x i ) M i, then, for some strictly positive µ i let { ui (x û i (x i, d i ) = i ) + µ i d i, if x i S i u i (x i ), otherwise, for every (x i, d i ) X i. We will check that Assumption (H 2 ) is satisfied for every û i. We will make use of the following lemma, the proof of which is in the Appendix. Lemma 1. Let B be a compact, convex set of R l and x be in B. Let λ : B R + {+ } be defined by λ(x) = sup{β 0 : x + β(x x ) B}. Then λ is upper semicontinuous and strictly quasi-concave. To prove that û i is quasi-concave and upper semicontinuous, it suffices to prove that the set L α i = {(x i, d i ) X i R + : û i (x i, d i ) α} is closed and convex for every α. Case 1. It is clear that L α i = L α i R +, with L α i = {x i X i : u i (x i ) α}. Therefore, L α i is closed and convex for every α. Case 2. (a) If α M i, then obviously L α i = L α i R +. (b) If M i < α 1 + M i, then one can easily prove that L α i = σ i 1 ( ) (M i +1 α) R +, where σ i 1 ( ) = {x S 1 (M i +1 α) i : λ i (x) (M i }. From Lemma 1, this +1 α) set is closed and convex. (c) If 1 + M i < α, then obviously L α i =. Case 3. We follow here the proof given by Le Van and Minh (2004), in which we have two cases, (a) If α < M i. We claim that L α i = L α i R +. Indeed, let (x i, d i ) L α i. It follows that û i (x i, d i ) α, and there are two possibilities for x i : 7

If x i / S i, then û i (x i, d i ) = u i (x i ). It implies that u i (x i ) α or x i L α i, and hence (x i, d i ) L α i R +. If x i S i, then u i (x i ) = M i > α. Thus, it follows that x i L α i (x i, d i ) L α i R +. and So, L α i L α i R +. But it is obvious that L α i R + L α i. (b) If α M i. We claim that L { α i = S i d i d i α M i µ }. Indeed, if û i (x i, d i ) α, then x i S i. In this case, û i (x i, d i ) = M i + µd i α, and hence d i α M i µ. The converse is obvious. We have proved that û i is upper semicontinuous and quasi-concave, for every i I. Now, we prove that û i is strictly quasi-concave. Claim 1. The utility function û i is strictly quasi-concave, for every i I. Proof. See the Appendix. Obviously, the set of individually rational feasible allocations of economy Ê is compact since X i is compact for every i I. Assumption (H 3 ) is fulfilled by economy Ê. We now prove that the utility function û i has no satiation point on the set  i, the projection of  onto X i. Case 1. It is obvious. Case 2. Since λ i (x i ) = +, it suffices to prove that λ i (x i ) < +, for any x i A i. For that, take x i S i A i. If λ i (x i ) = +, then for all β 0, x i +β(x i x i ) S i. Since O + S i = {0}, one obtains x i = x i which contradicts x i A c i. Case 3. Indeed, let (x i, d i ) X i R +. Take any x i S i and d i > d i. We have û i (x i, d i) = u i (x i) + µd i > u i (x i ) + µd i û i (x i, d i ). We have proved that for any i I, û i has no satiation point. Summing up, Assumptions (H 1 ) (H 4 ) are fulfilled in the economy Ê. From Theorem 1, there exists a Walras quasi-equilibrium ((x i, d i ) i I, (p, q )) with (p, q ) (0, 0). It satisfies: 8

(i) i I (x i, d i ) = i I (e i, δ i ), (ii) for any i I, p x i + q d i = p e i + q δ i. Observe that the price q must be nonnegative. We claim that ((x i ) i I, p ) is a quasi-equilibrium with dividends (q δ i ) i I. Indeed, first, we have i I, p x i p e i + q δ i. Now, let x i X i such that u i (x i ) > u i (x i ). That implies x i / S i, and hence û i (x i, d i ) = u i (x i ). We have û i (x i, 0) = u i (x i ) and hence û i (x i, 0) > û i (x i, d i ). Applying the previous theorem, we obtain We have proved our proposition. p x i = p x i + q 0 p e i + q δ i. Now, we show that Proposition 1 still holds by dropping the assumption that every consumption set is compact and replacing it by the compactness of the utility set U. Our method of proving existence is new and will be useful for the proof of the main result. Theorem 2. Assume (H 1 ) (H 3 ). Then there exists a quasi-equilibrium with dividends. Proof. Let (δ i > 0) i I. Let B(0, n) denote the closed ball centered at the origin with radius n, where n is an integer. Choose N sufficiently large such that for all n > N we have e i B(0, n), for all i I. Consider the sequence of economies {E n } defined by E n = (Xi n, u i, e i ) i I where Xi n = X i B(0, n). Any economy E n satisfies the assumptions of Proposition 1. Let û n i : Xi n R + R be the modified utility associated with E n as in the proof of Proposition 1. The utility functions {(û n i ) i I } have no satiation points and are strictly quasi-concave. Hence, from Proposition 1 there exists a quasi-equilibrium ((xi n ) i I, p n ) with dividends (q n δ i ) i I, and (p n, q n ) = 1. We have for each n, there exist (d n i )i I such that : (a) i I (x n i, d n i ) = i I (e i, δ i ), 9

(b) p n x n i + q n d n i = p n e i + q n δ i, (c) if x i Xi n satisfies u i (x i ) > u i (x n i ), then we have p n x i p n e i +q n δ i. Now, let n go to +. Since U is compact, the sequence {(u i (x n i ) i I )} converges to (vi ) i I U. There exist (x i ) i I A which satisfy vi u i (x i ), for all i I. We can also assume that (p n, q n ) converge to (p, q ) (0, 0) and (d n i ) i I converges to (d i ) i I. We claim that ((x i ) i I, p ) is, for the initial economy E, a quasi-equilibrium with dividends (q δ i ) i I. Indeed, we have (a) i I (x i, d i ) = i I (e i, δ i ). Let us prove that we have p x i + q d i = p e i + q δ i. There exists N such that for all n > N, x i Xi n, for all i I. In the following we take n > N. Let (x i, d i ) Xi n R + which satisfies û n i (x i, d i ) > û n i (x i, d i ). Let θ ]0, 1[. We have û n i (θx i + (1 θ)x i, θd i + (1 θ)d i ) > û n i (x i, d i ). Thus, p n (θx i +(1 θ)x i )+q n (θd i +(1 θ)d i ) p n e i +q n δ i. Let θ converge to 0. We get for all i I, p n x i + q n d i p n e i + q n δ i. Since i I (x i, d i ) = i I (e i, δ i ), we have for all i I, p n x i + q n d i = p n e i + q n δ i. Letting n go to +, we obtain p x i + q d i = p e i + q δ i. Now, let u i (x i ) > u i (x i ). For n large enough, x i Xi n. Since u i (x n i ) converges to vi u i (x i ), for any n sufficiently large, we have u i (x i ) > u i (x n i ) and hence p n x i p n e i + q n δ i. When n goes to + we obtain p x i p e i + q δ i. We have proved that ((x i ) i I, p ) is, for the initial economy E, a quasi-equilibrium with dividends (q δ i ) i I. 3.2 New nonsatiation assumption The above concept of equilibrium with dividends is used in the literature whenever the standard nonsatiation assumption fails to be satisfied, that is to say, the satiation area overlaps with the individually feasible consumption set. The underlying idea is to allow the nonsatiated consumers to benefit, through dividends, from the budget surplus created by non budget-binding optimal consumptions of satiated consumers. A shortcoming of equilibrium with dividends is that, granting additional incomes to consumers could possibly be inconsistent with the spirit of decentralized markets. In the following we introduce our new nonsatiation assumption. First, observe that under (H 1 ) (H 3 ), for every i I, there exists x i A i which satisfies u i ( x i ) = M i = max{u i (x i ) x i A i }. 10

Using M i, Assumption (H 4 ) could be rewritten in another way. (H 4 ) For every i, there exists x i A c i such that u i (x i) > M i. We introduce a new nonsatiation condition (H 4) weaker than (H 4 ). (H 4) For every i, there exists x i A c i such that u i (x i) M i. Assumption (H 4) allows to have satiation points inside the individually feasible consumptions set, provided that the satiation area is not a subset of the individually feasible consumption set. We now state the main contribution of this paper. We demonstrate that using our new nonsatiation assumption (H 4) leads us to the existence of a Walras quasi-equilibrium. Hence, we show that the concept of (quasi- )equilibrium with dividends is relevant only when the satiation area is a subset of individually feasible consumption set. Theorem 3. Assume (H 1 ) (H 3 ) and (H 4). Then there exists a Walras quasi-equilibrium. Proof. Consider again the sequence of economies {E n } in the proof of Theorem 2. For any i I, from (H 4), one can take some x i A c i such that u i (x i) M i. Choose n large enough such that for all n > N, x i Xi n, for all i I. From the proof of Theorem 1, there exists a quasi-equilibrium ((xi n ) i I, p n ) with dividends (q n δ i ) i I, and (p n, q n ) = 1. For each n, there exist (d n i ) i I such that: (a) i I (x n i, d n i ) = i I (e i, δ i ), (b) p n x n i + q n d n i = p n e i + q n δ i, (c) and if x i Xi n satisfies u i (x i ) > u i (x n i ) then we have p n x i p n e i + q n δ i. Now, we show that q n d n i = 0. It is clear that from (H 4) we have just to consider only cases 1 and 2. Thus, let û n i (x i, 0) > û n i (x n i, d n ) = û n i (x n, 0). We then have, i p n x i p n e i + q n δ i = p n x n i + q n d i. For any λ ]0, 1[, from the strict quasi-concavity of û n i, we have û n i (λx i + (1 λ)x n i ) > û n i (x n i ) and hence p n (λx i + (1 λ)x n i ) p n x n i + q n d n i. Letting λ converge to zero, we obtain q n d n i 0. Thus, q n d n i = 0. Since = i I δ i > 0, it follows that q n = 0. In this case, one deduces i I d n i 11 i

p n = 1 and therefore ((x n i ) i I, p n ) is a Walras quasi-equilibrium for E n. Finally, let n go to +. Since U is compact, the sequence {(u i (x n i )) i I } converges to (v i ) i I and there exist (x i ) i I A which satisfy v i u i (x i ), for all i I. We also have p n converges to p 0. We claim that ((x i ) i I, p ) is a Walras quasi-equilibrium. Indeed, we have i I x i = i I e i. Now, let u i (x i ) > u i (x i ). For n large enough, x i Xi n. Since u i (x n i ) converges to v i u i (x i ), for any n sufficiently large, we have u i (x i ) > u i (x n i ) and hence p n x i p n e i. When n goes to +, we obtain p x i p e i. We have proved that ((x i ) i I, p ) is, for the initial economy E, a Walras quasi-equilibrium. 4 Securities market In securities markets with short-selling, Werner (1987) introduces a nonsatiation condition which requires each trader to have a useful portfolio. Accordingly, Werner (1987) proves the existence of a competitive equilibrium in securities markets. For each agent i I, we define the weakly preferred set at x i X i P i (x i ) = {x X i u i (x) u i (x i )}. Under assumptions (H 1 ) (H 2 ), the weak preferred set P i (x i ) is convex and closed for every x i X i. We define the i th agent s arbitrage cone at x i X i as O + Pi (x i ), the recession cone of the weakly preferred set P i (x i ). Also, we define the lineality set L i (x i ) as the largest subspace contained in the arbitrage cone O + Pi (x i ). For notational simplicity, we denote each agent s arbitrage cone and lineality space at endowments in a special way. In particular, we will let, R i := O + Pi (e i ), and L i := L(e i ). Werner (1987) assumes the two following assumptions: [W1] (Uniformity) O + Pi (x i ) = R i, for all, x i X i, for each i I. [W2] (Werner s nonsatiation) R i \ L i, for each i I. The first assumption asserts that every agent has uniform arbitrage cones. The second assumption is viewed as a nonsatiation assumption. It requires that there exists a useful net trade vector r i R i \ L i. This is a portfolio which, when added, at any rate, to any given portfolio increases the trader s utility. 12

Werner (1987) also introduces a no-arbitrage condition [WNAC], which stipulates that there exists a price system at which the value of all useful portfolios is positive. [WNAC] The economy E satisfies I i=1 SW i, where S W i = {p R l p y > 0, y R i \ L i } is Werner s cone of no-arbitrage prices. It follows from Allouch, Le Van and Page (2002) that [WNAC] implies that U is compact. Hence, Assumption (H 3 ) is satisfied. In the following proposition we show that Werner s nonsatiation implies assumption (H 4). Proposition 2. Assume (H 1 ) (H 2 ), [W1] and [WNAC]. Then, [W2] implies (H 4). Proof. Let i 0 I. Since [WNAC] implies that U is compact, there exists x i 0 = argmax Ai0 u i0 (.). Let v i0 R i0 \ L i0 and let {(λ n )} be a sequence of real numbers such λ n 0, for all n and λ n goes to +. Since v i0 is a useful direction, it follows that u i0 (x i 0 +λ n v i0 ) u i0 (x i 0 ) = M i0, for all n. We claim that, that there exists n 0, such that (x i 0 + λ n0 v i0 ) / A i0. Suppose not, then there exists {(x n i )} a sequence in A such that x n i 0 = x i 0 + λ n v i0, for all n. Since I i=1 xn i goes to +, without loss of generality, we can assume that for all i I, x n i I i=1 xn i v i, such that for all i I \{i 0 }, v i R i and v i0 + i I\{i 0 } v i = 0. From [WNAC] there exists p I i=1 SW i, then it follows that p (v i0 + v i ) = p 0 > 0, i I\{i 0 } which is a contradiction. Thus, (H 4) holds. Corollary 1. Assume (H 1 ) (H 2 ), [W1]-[W2] and [WNAC]. Then, there exists a Walras quasi-equilibrium. 13

4.1 Example Now, we provide an example where both the standard nonsatiation assumption (H 4 ) and Werner s nonsatiation [W2] fail to be satisfied. However, our new nonsatiation assumption (H 4) holds. In this example, we have the existence of a competitive equilibrium that could not have been inferred from standard existence theorems. Example Consider the economy with two consumers and two commodities. Consumer 1 has the following characteristics: X 1 = [0, 10] [0, 10], u 1 (x 1, y 1 ) = { min{x1, y 1 }, if either x 1 [0, 3] or y 1 [0, 3], 3 otherwise. e 1 = (6, 2). Consumer 2 has the following characteristics: X 2 = R 2 +, u 2 (x 2, y 2 ) = x 2 + y 2, e 2 = (2, 6). We have u 1 (e 1 ) = 2, u 2 (e 2 ) = 8. The satiation set of agent 1 is S 1 = [3, 10] [3, 10]. Let ζ 1 = (x 1, y 1 ), ζ 2 = (x 2, y 2 ). The set of individually rational feasible allocations is: A = {(ζ 1, ζ 2 ) X 1 X 2 : ζ 1 + ζ 2 = (8, 8), and u 1 (ζ 1 ) 2, u 2 (ζ 2 ) 8}. It is easy to see that (3, 3) A 1 and u 1 (3, 3) = 3 = M 1. Hence, for agent 1, Assumption (H 4 ) is not satisfied. It is also worth noticing that Werner s nonsatiation is not satisfied by agent 1, since X 1 is a compact set, and therefore R 1 = {0}. However, it is obvious that Assumption (H 4) is satisfied by both consumers, since (10, 10) / A 1 and u 1 (10, 10) = 3 = M 1, for agent 1 and (H 4 ) is satisfied for agent 2. One can easily show that the allocation ((4, 4), (4, 4)) together with the price (1, 1) is an equilibrium for the economy. 14

5 Appendix Lemma 1 Let B be a compact, convex set of R l and x be in B. Let λ : B R + {+ } be defined by λ(x) = sup{β 0 : x + β(x x ) B}. Then λ is upper semicontinuous and strictly quasi-concave. Proof. First, observe that λ(x) 1, x B. (i) Upper semicontinuity of λ: First we have: λ(x) = + x = x. Let x n converge to x. If lim sup λ(x n ) = +, take a subsequence which satisfies lim λ(x ν ) = +. Let ε > 0 be any small positive number. We have or x + (λ(x ν ) ε)(x ν x ) = z ν B, ν, (x ν x ) = zν x λ(x ν ) ε. Let ν go to +. We have zν x λ(x ν ) ε goes to 0, hence, xν converges to x and x = x. That implies λ(x) = + and lim sup λ(x n ) = λ(x). Now, assume lim sup λ(x n ) = A < +. Without loss of generality, one can assume that λ(x n ) converges to A. Take any ε > 0 small enough. Then, n, x +(λ(x n ) ε)(x n x ) B. Let n go to infinity. Then x +(A ε)(x x ) B. That implies λ(x) A ε for any ε > 0 small enough. In other words, λ(x) lim sup λ(x n ). We have proved that λ is upper semicontinuous. (ii) Quasi-concavity of λ: Let x 1 B, x 2 B, θ [0, 1] and x = θx 1 + (1 θ)x 2. Assume λ(x 1 ) λ(x 2 ). As before take any ε > 0 small enough. We then have x + (λ(x 1 ) ε)(x 1 x ) B, and x + (λ(x 1 ) ε)(x 2 x ) B, 15

since λ(x 1 ) λ(x 2 ). Thus θ(x + (λ(x 1 ) ε)(x 1 x )) + (1 θ)(x + (λ(x 1 ) ε)(x 2 x )) B since B is convex. We obtain x +(λ(x 1 ) ε)(θx 1 +(1 θ)x 2 x ) B. Hence λ(θx 1 + (1 θ)x 2 ) λ(x 1 ) ε for any ε > 0 small enough. In other words, λ(θx 1 + (1 θ)x 2 ) min{λ(x 1 ), λ(x 2 )}. We have proved the quasi-concavity of λ. (iii) Strict quasi-concavity of λ: Let λ(x 2 ) > λ(x 1 ). We first claim that x + λ(x 1 )(x 1 x ) B. Indeed, n, x + (λ(x 1 ) 1 n )(x 1 x ) B. Let n go to +. The closedness of B implies that x + λ(x 1 )(x 1 x ) B. Now, let θ ]0, 1[. We claim that λ(θx 1 + (1 θ)x 2 ) > λ(x 1 ). For notational simplicity, we write λ 1 = λ(x 1 ), λ 2 = λ(x 2 ). Let A satisfy λ 1 < A < λ 2. Then x + A(x 2 x ) B. Hence θa λ 1 (1 θ) + θa (x + λ 1 (x 1 x λ 1 (1 θ) )) + λ 1 (1 θ) + θa (x + A(x 2 x )) B, or equivalently x + Thus, λ(θx 1 + (1 θ)x 2 ) Aλ 1 λ 1 (1 θ) + θa (θx 1 + (1 θ)x 2 x ) B. Aλ 1 λ 1 (1 θ)+θa > λ 1, since A > λ 1. Claim 1 The utility function û i is strictly quasi-concave. Proof. Let and θ ]0, 1[. We claim that û i (x 2, d 2 ) > û i (x 1, d 1 ) (1) û i (θ(x 2, d 2 ) + (1 θ)(x 1, d 1 )) > û i (x 1, d 1 ). (2) Let us distinguish the three cases again: Case 1. The claim is obviously true since (1) is equivalent to u i (x 2 ) > u i (x 1 ) and u i is assumed to be strictly quasi-concave. Case 2. We have two sub-cases. 16

(a) If x 1 S i, then x 2 S i. In this case, (1) is equivalent to λ i (x 2 ) > λ i (x 1 ). From Lemma 1, λ i is strictly quasi-concave. Hence (2) is true, since S i is convex, and therefore θx 2 + (1 θ)x 1 S i. (b) If x 1 / S i. Then (1) is equivalent to u i (x 2 ) > u i (x 1 ). Since u i is a strictly quasi-concave function, we obtain Therefore, one has u i ((1 θ)x 1 + θx 2 ) > u i (x 1 ). û i (θ(x 2, d 2 ) + (1 θ)(x 1, d 1 )) u i (θx 2 + (1 θ)x 1 ) > u i (x 1 ) = û i (x 1, d 1 ), and consequently (2) is true. Case 3. We can consider the following cases: (a)if x 1 S i, then x 2 S i. Hence, we have û i (x 1, d 1 ) = M i + µd 1, û i (x 2, d 2 ) = M i + µd 2. It follows from (1) that d 2 > d 1. Hence, (1 θ)d 1 + θd 2 > (1 θ)d 1 + θd 1 = d 1. Since (1 θ)x 1 + θx 2 S i, we deduce û i ((1 θ)x 1 + θx 2, (1 θ)d 1 + θd 2 ) = M i + µ((1 θ)d 1 + θd 2 ) > M i + µd 1 = û i (x 1, d 1 ). (b)if x 1 / S i. Then (1) implies u i (x 2 ) > u i (x 1 ). Since u i is a strictly quasiconcave function, we obtain Then, it follows that u i ((1 θ)x 1 + θx 2 ) > u i (x 1 ). û i ((1 θ)x 1 +θx 2, (1 θ)d 1 +θd 2 ) u i ((1 θ)x 1 +θx 2 ) > u i (x 1 ) = û i (x 1, d 1 ). The proof of the claim is complete. 17

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